Average Error: 2.2 → 0.1
Time: 28.7s
Precision: 64
Internal Precision: 320
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 2.439311431159999 \cdot 10^{+130}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(k \cdot 10 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;(\left({\left(e^{m}\right)}^{\left(\log k\right)} \cdot \frac{a}{{k}^{4}}\right) \cdot 99 + \left(\left(\frac{a}{k} - \frac{10}{k} \cdot \frac{a}{k}\right) \cdot \frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{k}\right))_*\\ \end{array}\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Split input into 2 regimes

  2. if k < 2.439311431159999e+130

    1. Initial program 0.1

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]

    if 2.439311431159999e+130 < k

    1. Initial program 9.9

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]

    2. Taylor expanded around -inf 63.0

      \[\leadsto \color{blue}{\left(99 \cdot \frac{a \cdot e^{m \cdot \left(\log -1 - \log \left(\frac{-1}{k}\right)\right)}}{{k}^{4}} + \frac{a \cdot e^{m \cdot \left(\log -1 - \log \left(\frac{-1}{k}\right)\right)}}{{k}^{2}}\right) - 10 \cdot \frac{a \cdot e^{m \cdot \left(\log -1 - \log \left(\frac{-1}{k}\right)\right)}}{{k}^{3}}}\]

    3. Simplified0.1

      \[\leadsto \color{blue}{(\left({\left(e^{m}\right)}^{\left(\log k\right)} \cdot \frac{a}{{k}^{4}}\right) \cdot 99 + \left(\frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{k} \cdot \left(\frac{a}{k} - \frac{10}{k} \cdot \frac{a}{k}\right)\right))_*}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 2.439311431159999 \cdot 10^{+130}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(k \cdot 10 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;(\left({\left(e^{m}\right)}^{\left(\log k\right)} \cdot \frac{a}{{k}^{4}}\right) \cdot 99 + \left(\left(\frac{a}{k} - \frac{10}{k} \cdot \frac{a}{k}\right) \cdot \frac{{\left(e^{m}\right)}^{\left(\log k\right)}}{k}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 28.7s)Debug logProfile

herbie shell --seed 2018216 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))